
We studied in this work the kinetic EOS (both with and without the HMT) in some details in the general dimension \(d\) , the next important step is to effectively/reasonably incorporate the nucleon-nucleon interactions/potentials in a self-consistent way, either at the phenomenological or at the microscopic levels. For example, one needs to consider how to determine the model parameters in \(d\) D, and the basic principles and/or symmetries to be considered, etc.
Before detailed schemes (fitting schemes and/or the principles) are available, the \(\epsilon \) -expansion provides a practical starting point, e.g., one can use (REF ) to investigate how the 3D knowledge on the symmetry energy could induce relevant information in dimensions \(\epsilon +d_{\mathrm {f}}\) .

Considering the HMT issue, how can one obtain the EOS of ANM based on the HVH theorem in general \(d\) D, i.e., the close relation between the partial derivative of the energy density with respect to the density and the single energy at \(k_{\mathrm {F}}\) , since when the momentum distribution has depletion as well as the HMT, one does not have \(\partial \varepsilon /\partial \rho =e(k_{\mathrm {F}})\) , where \(e\)  is the single-nucleon energy as the sum of the kinetic part and the potential part, and \(\varepsilon \)  is the energy density of the system, i.e., \(e(k_{\mathrm {F}})\)  is generally not the chemical potential of the nucleons (which however holds for the FFG model). This is in fact a fundamental problem in (nuclear) many-body theories [1]}.

As discussed in the last section, considering that the many-nucleon system in low dimensions is close to its FFG counterpart, can one rely on it (e.g., via the higher-order \(\epsilon \) -expansions) to extract useful information for the corresponding EOS in 3D?
A related issue is on the correlation between the symmetry energy and its slope parameter. It is known that the latter (at \(\rho _0\) ) in 3D is constrained to be about \(L\approx fS\)  with \(f\)  being about 2-3 [2]} and \(S\equiv E_{\mathrm {sym}}(\rho _0)\) , which is in fact quite close to its FFG prediction \(L=3S\) .
If the low-dimensional system is close to the free model, can one establish the corresponding correlation (even with the effective potential included) to be near \(L\approx 3S\)  in 2D or \(L\approx 6S\)  in 1D, or more generally \(L\approx 6S/d\) ?

In this work, we study the EOS of ANM in a space of general dimension \(d\)  totally adopting the non-relativistic calculations. Since the relativistic kinematics, the nucleon-nucleon interactions, and/or the dimensionality together may effectively affect the EOS of ANM [3]}, it would be extremely interesting to combine the relativistic kinematics and the fruitful nucleon-nucleon interactions (we have already found that the momentum dependence of the single-nucleon potential could effectively influence certain problems) in the general dimension \(d\)  and to explore how they influence each other coherently.
See the following overall scheme (see (REF ) for the definition of \(\theta _{\mathrm {f}}\)  and \(e_J\)  is the general relativistic single-nucleon energy [4]}) on its inner relations.
\(\boxed{{\begin{matrix}\begin{array}{c}e_J(\rho ,\delta ,|\mathbf {k}|)~\mbox{in~$d$D}\\\left(\hspace*{-4.26773pt}\begin{array}{c}\mathrm {NN~interactions}\\\mathrm {dimensionality}\end{array}\hspace*{-4.26773pt}\right)\end{array}&\xrightarrow[\mathrm {HVH/others}]{\delta \approx 0}&\begin{array}{l}\mbox{Relativistic}\\\mathrm {characteristics:}\\E_{\mathrm {sym}}(\rho ),L(\rho ),\cdots \end{array}&\xrightarrow[\epsilon =d-d_{\mathrm {f}}]{|\mathbf {k}|,k_{\mathrm {F}}\ll M}&\left\lbrace \hspace*{-4.26773pt}\begin{array}{l}\mathrm {NR~formulas}\\\mbox{$d_{\mathrm {f}}$D~quantities}\end{array}\right.\\\mathbox{mphantom}{\scriptstyle |\mathbf {k}|,k_{\mathrm {F}}\ll M}\downarrow {\scriptstyle |\mathbf {k}|,k_{\mathrm {F}}\ll M}&& {\scriptstyle \theta _{\mathrm {f}}\approx 0}\downarrow \mathbox{mphantom}{\scriptstyle \theta _{\mathrm {f}}\approx 0}&& \\\begin{array}{c}|\mathbf {k}|^2/2M+U_J(\rho ,\delta ,|\mathbf {k}|)\\(\mathrm {nucleon~potential})\end{array}&&\begin{array}{l}\mathrm {Characteristics:}\\P(\rho ),K_0(\rho ),J_0(\rho ),\cdots \end{array}\\\downarrow && \mathbox{mphantom}{\scriptstyle \epsilon =d-d_{\mathrm {f}}}{\scriptstyle  {|\mathbf {k}|,k_{\mathrm {F}}\ll M}}\downarrow {\scriptstyle \epsilon =d-d_{\mathrm {f}}}\mathbox{mphantom}{\scriptstyle  {|\mathbf {k}|,k_{\mathrm {F}}\ll M}}&& \\\mathrm {NR~formulas}&\xrightarrow[ \epsilon =d-d_{\mathrm {f}}\approx 0]{\delta \approx 0,\theta _{\mathrm {f}}\approx 0}&\begin{array}{l}\mbox{NR~formulas~in $d_{\mathrm {f}}$D:}\\P(\rho ),K_0(\rho ),J_0(\rho ),\cdots \end{array}\\\end{matrix}}}\) 

The role played by the momentum-dependences of the single-nucleon potentials especially that of the symmetry potential \(U_{\mathrm {sym}}\)  on the symmetry energy is revealed. Currently, we can not determine whether an enhancement or a reduction may be induced to the \(E_{\mathrm {sym}}(\rho )\)  if \(d\)  is perturbed, see the formula () and Tab. REF , the discussions after Eq. (REF ) and those after Eq. (REF ).
To be deterministic, more detailed information about the momentum dependence of the symmetry potential \(U_{\mathrm {sym}}\)  and more accurate calculations (e.g., to higher orders of \(\epsilon \) ) are needed.

The last but not least, it is really important to find applications where the low-dimensional EOS of ANM could be used. As we have speculated in the Introduction, the crust of a neutron star may be treated as a quasi-2D system while its non-rotating core may be consider as a 1D system, and certain sub-systems of particles in heavy-ion reactions may be also treated as a low-dimensional manifold. Finding novel signatures of these problems adopting the general \(d\) D EOS of ANM is a big challenge and also an exciting task. For example, the radial equation relating the crust EOS of neutron star matter with observables should also be derived starting from a suitable 2D setting [5]}, [6]} (e.g., via the 2D Christoffel coefficients). Thus, the conventional Tolman-Oppenheimer-Volkoff equation [7]} and its input EOS may both need to be modified to account for the changes from 3D to \(d\) D spaces.

